microwave background anisotropies and large scale ...by linde and mukhanov. we provide constraints...

Microwave background anisotropies and large scale structure constraintson isocurvature modes in a two-field model of inflation

Elena PierpaoliDepartment of Physics and Astronomy, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada

Juan Garcıa-BellidoTheoretical Physics, Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, U.K.

Stefano BorganiINFN Sezione di Trieste, c/o Dipartimento di Astronomia dell’Universita, via Tiepolo 11, I-34131 Trieste, Italy

INFN Sezione di Perugia, c/o Dipartimento di Fisica dell’Universita, Via A. Pascoli, I-06123 Perugia, Italy(November 17, 1999)

In this paper we study the isocurvature mode contribution to the cosmic microwave backgroundanisotropies and the large scale structure power spectrum, for a two-field model of inflation proposedby Linde and Mukhanov. We provide constraints on the parameters of the model by comparing itspredictions with observations of the microwave background anisotropies, large scale structure dataon the galaxy power spectrum, and the number density of nearby galaxy clusters. We find thatsuch models are consistent with observations for a narrow range of parameters. As our main result,we find that only a very small isocurvature component is allowed, α ≤ 0.006, for any underlyingFriedmann model. Furthermore, we give the expected resolution with which the model parameterswill be determined from future satellite missions like MAP and Planck, for a fiducial flat ΛCDMmodel. We find that Planck mission will be able to detect such small contributions, especially ifpolarization information is included. The isocurvature spectral index niso will also be determinedwith better than 8% precision.

PACS number: 98.80.Cq Preprint IMPERIAL-TP-98/99-72, UBC-COS-99-04, hep-ph/9909420

I. INTRODUCTION

The future satellite experiments MAP [1] and Planck[2] open an exciting era for cosmologists and for particlephysicists. The high resolution and sensitivity of theseexperiments will allow such a precise determination ofthe cosmic microwave background (CMB) power spec-trum of temperature and polarization anisotropies thatit will soon be conceivable to test different cosmologi-cal models with great accuracy [3]. Meanwhile, galaxysurveys, like 2dF and Sloan Digital Sky Survey (SDSS),aimed at measuring several hundred of thousand redshiftswill provide a map of the Universe with unprecedentedprecision and extension. Any viable cosmological modelmust produce reasonable fits to both the observed CMBand large scale structure (LSS) power spectra, and mustbe tested on the basis of all available data.

Within the context of inflationary scenarios, Gaussianadiabatic fluctuations are often assumed as a standardprediction. However, besides the usual adiabatic fluc-tuations, other independent modes may be present, e.g.isocurvature fluctuations [4–7], and in many cases theyare non-Gaussian [8]. In particular, the isocurvaturemode is an entropy perturbation, characterized by anappropriate balance of the fluctuations in the differentcomponents, such that the spatial curvature remains un-perturbed. There is an attractive model that has been

proposed some time ago that considers pure isocurva-ture perturbations in the baryon component [9], but un-fortunately seems to be ruled out by present observa-tions [10,11]. Most of the recent models are actually colddark matter (CDM) isocurvature models [12], but thereis also an ingenious neutrino isocurvature model [13].

Far from being academic, the reason for consideringalso isocurvature fluctuations resides in the fact thatmany different inflationary models, with more than onescalar field, predict the formation of significant isocurva-ture fluctuations during the inflationary era [4,14]. As fortheir predictions on the density perturbation power spec-tra, they may differ among themselves in having differentamplitudes and tilts of isocurvature and adiabatic spec-tra, as well as for the statistical nature of both modes.Many inflationary models predict that both the isocurva-ture and adiabatic fluctuations have a nearly scale invari-ant (Harrison–Zeldovich) spectrum with Gaussian statis-tics. These models have been tested against LSS in therecent literature [15,16], which showed that only a smallfraction of isocurvature component seems to be allowedby present observations.

In this paper we focus on a particular inflationary sce-nario proposed by Linde and Mukhanov [17], in whichnone of the two conditions mentioned above is neces-sarily satisfied. In fact, here the isocurvature fluctua-tions are non–Gaussian, more specifically, they are χ2

1

distributed,1 and their spectrum is a power law, withspectral index niso > 1. We test this model with thepresent observations on CMB and LSS, and we also makepredictions on how well the future satellite experimentswill be able to measure the relevant model parameters.

In previous papers [15,16,18] the mixed spectra ofadiabatic plus isocurvature modes were assumed to beindependent, Gaussian and approximately scale invari-ant. Some [15,16] used the large scale structura data,together with the COBE-DMR normalization, to con-strain the models, while others [18] only used the CMBanisotropies, with or without polarization. In our pa-per, we have combined both CMB and LSS observations,for a non scale invariant spectrum of perturbations, andincluded also the gravitational wave contribution.

In section II we review how isocurvature modesmay arise from inflation, and we describe the Linde–Mukhanov model which inspired this work. In section IIIwe introduce the power spectra for matter and radiationin the mixed (adiabatic + isocurvature) case, showing theeffect of the amplitude and tilt of the different isocurva-ture contributions. In section IV we show the compar-ison of the mixed spectra with the available data: weconstrain the parameter range considering a joint analy-sis of both CMB and LSS, and we estimate with whichprecision the future satellite experiments will determinethe relevant parameters. Finally, section V is dedicatedto a general discussion of the results.

II. ISOCURVATURE MODES FROM INFLATION

Isocurvature perturbations are generated during infla-tion whenever there is more than one scalar field present.They correspond to entropy perturbations that do notperturb the metric, and thus the spatial curvature. Theytypically arise when one of the fields is fixed by its poten-tial during inflation, the inflaton energy later decays intorelativistic particles and redshifts away, while the otherfield’s energy becomes the dominant contribution. De-pending on model parameters, the relative contributionof adiabatic to isocurvature perturbations may be notice-able in the microwave background anisotropies and largescale structure, and they may have in principle very dif-ferent spectral tilts, e.g. blue (n > 1) or red (n < 1).Furthermore, depending on the evolution during infla-tion, the statistics of the different components (adiabaticand isocurvature) could be very different, e.g. Gaussianand χ2 distributed, respectively. Such a complicated phe-nomenology requires a detail analysis in order to confirmwhether a particular model is ruled out by observations.

1However, slight variants of the model can also give a Gaus-sian isocurvature mode [17].

In this paper we will concentrate in a particular re-alization of a mixed adiabatic and isocurvature modelproposed recently by Linde and Mukhanov.

A. The Linde–Mukhanov model

The model of Ref. [17] is probably the simplest one canthink of that produces isocurvature perturbations duringinflation. It has two coupled massive scalar fields de-scribed by the scalar potential,

V (φ, σ) =12M2φ2 +

12m2σ2 +

12g2σ2φ2 . (1)

In principle, inflation could occur along either of the val-leys at φ = 0 or σ = 0, depending on initial conditions. Ina chaotic inflation approach one expects that the fieldswill start at very large values, φ, σ MP , where thecoupling term g2σ2φ2 dominates. Let us suppose thatinitially one of the fields has a larger value, say |φ| > |σ|and thus the field σ rapidly settles at σ = 0. Then infla-tion occurs along the σ = 0 valley, with energy density12M2φ2 and a Hubble constant

H2 =4πM2

3M2P

φ2 . (2)

During inflation the mass of the σ field becomes

m2 = m2 + ν H2 (3)

where ν = 3g2M2P /4πM2 is a constant. Typically, during

inflation the second term dominates and thus the modelgives a mass term of the σ field proportional to the rateof expansion.

The quasi-de-Sitter evolution during inflation providesa neat way to generate metric perturbations from quan-tum fluctuations. Those of the inflaton will give rise toadiabatic density perturbations, since the energy den-sity during inflation is proportional to the inflaton fieldfluctuations, δρ ∼ V ′(φ) δφ. On the other hand, quan-tum fluctuations of the σ field will not generate curvatureperturbations since the inflationary trajectory lies along〈σ〉 = 0. Nevertheless, after inflation the energy den-sity of the σ field may come to dominate the evolutionof the universe (e.g. as a cold dark matter component)and its fluctuations would then contribute as isocurva-ture perturbations [4]. Let us compute the amplitude ofthose perturbations. For a massive field with m2 H2

during inflation, the amplitude of the long wavelengthperturbation of the σ field at the end of inflation is givenby

k3|σ2k| =

H2

2

(k

H

)2m2/3H2

, (4)

and the average perturbations of energy density in the σfield, δρσ = m2(δσ)2/2 can be estimated as [17]

2

δρσ

ρtot∼ m2

M2P

(k

H

)2m2/3H2

, (5)

which corresponds to a “blue” spectral index

niso ≈ 1 + 4m2/3H2 ' 1 + 4ν/3 > 1 . (6)

The small ratio m2/M2P ensures that the σ field does not

contribute initially to the perturbations of the metric,i.e. it generates isocurvature perturbations, which muchlater could end up dominating, as mentioned above. Thedetailed evolution is very model dependent [17]. We willassume, following Linde and Mukhanov, that the correc-tion νH2 to the mass of the σ field disappears soon afterinflation, when the inflaton field φ decays into relativisticparticles while the σ field remains stable or decays verylate, and thus its energy density (in coherent oscillationsof the field) may dominate in the form of cold dark mat-ter today. Under these assumptions one can estimate thecorresponding density contrast [17]

δρσ

ρσ∼ C(k)

(k

M

)2ν/3

, (7)

for k min[M√

m/M, M exp(−1/2ν)], where

C(k) =√

ν

ln(M/m) + 3/ν

[ln

(M

k

)]1−ν/3(M

m

)2ν/3

.

(8)

This isocurvature perturbation has non-Gaussian statis-tics, in fact a χ2 distribution, because it arises from thesquare of a Gaussian field [see Eq. (5)]. 2

On the other hand, the adiabatic density perturba-tions generated by the inflaton field φ during inflationcontribute with a Gaussian spectrum with amplitude

δρk

ρ'

√43π

NkM

MP

(k

aH

)−2/Nk

, (9)

where Nk = Nhor = 65 is the number of e-folds beforethe end of inflation when the mode with wavenumber kcorresponding to our present horizon crossed the Hubblescale during inflation, with the spectral index

nad = 1− 2Nk

. (10)

There is enough freedom in the model parameters(ν, M, m) to have the Gaussian adiabatic perturbations

2One could avoid having non-Gaussian statistics for theisocurvature component by giving the σ field a vacuum ex-pectation value, 〈σ〉 6= 0. In this way, the main isocurva-ture contribution would come from the term δρσ = m2〈σ〉δσ,which is Gaussian distributed.

dominate the spectrum on large scales, while the non-Gaussian isocurvature perturbations dominate on smallscales. In practice, the latter will never dominate atscales of cosmological interest since, as we shall illustratein Sections IV and V, the relative contribution of theisocurvature component is constrained by present obser-vations to be very small.

We will thus consider a mixed model of adiabatic andisocurvature perturbations arising from inflation and con-tributing simultaneously to the temperature anisotropiesof the microwave background and the power spectrum oflarge scale structure. The relative tilts and amplitudesof the two spectra will allow us to compare with presentobservations and determine the likelihood functions for agiven set of model parameters.

We will use the CMBFAST code [19,20] to normalizethe model to COBE data [21–23], compute the Cl andpower spectra P (k), and then compare with observations.

Therefore, the post–inflationary power spectra for theindependent adiabatic and isocurvature components read

Pad(k) = 25× 10−10

(k

khor

)nad

, (11)

Piso(k) = d10 × 10−10

(k

kgal

)niso

, (12)

where d10 = O(1− 20) is an arbitrary normalization fac-tor, to be fixed by observations. We will discuss in thenext Section the relative normalization of the two com-ponents for a mixed power spectrum.

III. MATTER AND RADIATION POWERSPECTRA

In order to compare the theory with the data, we com-pute the power spectra of matter and radiation at thepresent epoch. For this purpose, we solve Boltzmannequations using CMBFAST, starting from a very earlyepoch (z ' 107) up to the present time.

In the presence of only one mode of fluctuations theactual matter power spectrum is:

P (k) = T 2(k)Pin(k) , (13)

where Pin(k) denotes the initial power spectrum and thetransfer function T (k) contains the information on howthe spectrum is modified through the evolution. T (k)is typically of order 1 on large scales (small k) and itbecomes less than 1 at smaller scales (larger k), the ac-tual shape depending on the specific mode considered(i.e., whether adiabatic or isocurvature), the cosmologi-cal parameters and on the nature of the dark matter con-tent. In our analysis, while we will allow for both adia-batic and isocurvature fluctuations for general Friedmannbackground, we will assume the dark matter content tobe contributed only by CDM and baryons, ignoring any

3

FIG. 1. Transfer function for the adiabatic (solid curve)and isocurvature (dashed curve) modes in the standard colddark matter model (Ωm = 1, h = 0.5 and ΩB = 0.05). Theisocurvature mode is more damped on small scales.

contribution that may come from neutrinos in the formof hot dark matter.

As for the adiabatic and isocurvature transfer func-tions, we take the CDM expressions

Tad(k) =ln(1 + 2.34q)

2.34q×[

1 + 3.89q + (16.1q)2 + (5.46q)3 + (6.71q)4]−1/4

; (14)

Tiso(k) =[1 +

(40q)2

1 + 215q + (16q)2(1 + 0.5q)−1+ (5.6q)8/5

]−5/4

, (15)

as provided by Bardeen et al. [24], where q = k/Γ(h−1Mpc)−1. Here Γ is the so called “shape” parameterand for CDM model is defined as Γ = hΩm exp(−ΩB −√

2hΩB/Ωm), so as to account for the small–scale damp-ing due to the presence of a non–negligible baryon frac-tion [25]. Note that we have redefined Eq. (15) withrespect to the expression given in Ref. [24] so as to havethe same small–k asymptotic behavior, T (k) → 1, forboth components. Among other things, this means thatniso = 1 corresponds to a scale invariant spectrum, in-stead of the usual niso = −3 appearing in the literature.We verified that Eqs. (14) and (15) reproduce quite wellthe outputs of the CMBFAST code for the k–range ofinterest for our analysis.

As an example, we show in Figure 1 the shape of thesetwo transfer functions for a particular choice of the cos-mological parameters. It is apparent that the isocurva-

FIG. 2. Power spectra of pure isocurvature (dotted curve)and adiabatic (dashed curve) modes with spectral indicesnad = 0.962 and niso = 1.56. The middle (solid curve) powerspectrum is a mixed one, corresponding to α = 8×10−3. Thedata points are from the APM survey.

ture case has a larger damping at large k values, due tothe fact that they start out as zero-curvature or isother-mal fluctuations and takes longer than the adiabatic onesto build up, after the matter–radiation equality epoch.

Since adiabatic and isocurvature are independentmodes of fluctuations, when both modes are present atthe same time the total power spectra of matter and radi-ation can be computed as a linear combination of the pureisocurvature and adiabatic ones. In the LM model con-sidered in this work, both the adiabatic and isocurvaturemodes have power law initial power spectra, but with dif-ferent spectral indices. Therefore, the overall spectrumcan be casted in the form

Pmix(k) = Nmix

[(kτ0)nadT 2

ad(k) + α (kτ0)nisoT 2iso(k)

],

(16)

where τ0 is the conformal time at present and α is a di-mensionless parameter that indicates the relative contri-bution of isocurvature and adiabatic perturbations (pureadiabatic and isocurvature power spectra correspond toα = 0 and α = ∞ respectively). The comparison of our αwith the mixing coefficients introduced by other authors(e.g. [15,16,18]) may not be straightforward because, incontrast with their approach, we have also allowed fornad 6= niso 6= 1. In the case nad = niso = 1, our α com-pares with the other choices as follows: α = (1−αS)/αS

[15], α = αKKSY [16], α = αEK/(1 − αEK) [18]. In

4

FIG. 3. Mixed CMB spectra with fixed niso value. Thesolid lines correspond to the pure adiabatic and isocurvaturemodes, normalized to COBE. The dotted–dashed lines cor-responds to α = 3, 6, 9, 12 × 10−3 from top to bottom. Thepoints shown here are the binned spectrum from Ref. [27].However, we have used the whole set of experiments fromTable I in the CMB analysis.

eq.(16), Nmix is the normalization factor, that we de-rived by normalizing the CMB spectrum to the COBEdata [22,23]. Different mixing coefficients α lead to dif-ferent normalizations:

Nmix =Nad

1 + αf. (17)

In the above expression, f = Nad/Niso where Nad

and Niso are the normalizations of the pure adiabaticand isocurvature scalar modes, and their ratio conveysthe information about the different Sachs–Wolfe con-tribution of isocurvature and adiabatic modes to CMBanisotropies. In the normalization, we also took into ac-count the tensor contribution to the anisotropies, accord-ing to the parameter specified below. A typical hybridspectrum is plotted in fig. 2, together with the APMdata points [26]. Given the big tilt of the isocurvaturepower spectrum, the mixed one has a slope similar tothe adiabatic one at small k. The effect of the differ-ent normalization is evident even for low values of the αcoefficient.

As for the CMB anisotropies, following the standardnotation, we describe their dependence on the directionn as

∆T (n)To

=∞∑l=0

l∑m=−l

alm Ylm(n) . (18)

FIG. 4. Mixed Cl spectra with different tilts of the isocur-vature component with fixed ratio α = 0.012. The solid lineis the pure adiabatic mode (nad = 0.962), the dotted line cor-responds to mixed isocurvature (α = 0.012) with niso = 1.36and the dashed line to niso = 1.61.

Therefore, the radiation power spectrum is defined as

Cl = 〈|alm|2〉 , (19)

where the brackets denote the ensemble average over dif-ferent realizations. When both scalar and tensor modesare present, the Cl can be decomposed as Cl = CS

l +CTl .

In Linde–Mukhanov model the tensor contribution is in-cluded in the adiabatic mode.

Similarly to what happens to the matter power spec-trum, the radiation power spectrum for the mixed casecan be found as a linear combination of the two indepen-dent adiabatic and isocurvature power spectra Cad

l andC iso

l :

Cmixl =

Cadl + fαC iso

l

1 + fα. (20)

In the expression above Cadl and C iso

l are both normalizedto COBE, according to ref. [23]. In computing the powerspectra, we modified the CMBFAST code [19,20] to ourpurposes. In figures 3 and 4 we show some examples ofCMB spectra, together with the estimated binned spec-trum from Bond et al. [27], which provides a visual indi-cation of the experimental status. However, we have usedthe whole set of experiments from Table I in the CMBanalysis. The effect of adding an isocurvature compo-nent is to add a lot of power on small multipoles throughthe SW effect. The anti–tilt of the spectrum is in gen-eral not enough to compensate this effect, and the first

5

FIG. 5. Typical values of ν as a function of M/m, with d10

in the range 1–10, from top to bottom.

acoustic peak is consequently lower than in the corre-sponding pure adiabatic case. Fig. 3 shows examples ofmixed spectra with different α values for fixed niso andnad. It shows that even a small α can cause a signifi-cant damping of the acoustic peaks. Moreover note thata high niso component also leads to suppressed acousticpeaks, for a fixed α value (see Fig. 4). This is becausethe bigger is niso the smaller is the normalization coef-ficient Niso and the bigger is f . Therefore for fixed α,the isocurvature Cl spectrum takes more weight in themixture.

A. Choice of model parameters

We will consider here a set of values for the parametersof the model. Let us start with the scalar spectrum. Theadiabatic tilt is given by Eq. (10),

nad = 1− 2/Nhor = 0.9692 , (21)

for Nhor = 65, while the spectral index for isocurvaturefluctuations is

niso = 0.9722 + 4ν/3 . (22)

Figure 5 shows some typical values of the parameter ν asa function of M/m, for different choices of d10.

As for the tensor contribution, we considered:

nT = −1/Nhor = −0.0154 , (23)

and a tensor to scalar ratio given by:

R =CT

2

CS2

' −7nT = 0.1077 , (24)

which is not negligible. In Fig. 4 the tensor contributionis very small and is already included in the solid line for

FIG. 6. Typical values of α as a function of the tilt param-eter ν in the isocurvature spectrum. The different curves referto different d10 ranging from 1 to 20, from bottom to top.

the adiabatic component, in order to make emphasis onits difference with respect to the isocurvature component.

With the above spectral properties, we determined therelative amplitude α in (16) to be related to the inputparameters d10 and ν,

α = d10 × 10−1.291−4.343 ν . (25)

For any value of d10 between 1 and 10, the tilt parameterν ranges between 0.29 and 0.48. As a consequence, niso

is found to be between 1.36 and 1.61.3 Note that while νfixes the value of the isocurvature spectral index, severalα values are still possible, depending on the d10 values.In fig. 6 we plot the value of α as a function of ν, for d10

in the range 1 − 20. In any case, the values of α foundare small, and for d10 < 32, α never exceeds 0.09. Inthe comparison with the data, we considered α < 0.08and 1.36 < niso < 1.61, and treated them as independentparameters, although they are not really independent [seeEq. (25)].

Note that we have represented in Fig. 6 the relativecontribution α of the isocurvature component to the to-tal power spectrum at COBE scales, i.e. large scales. Dueto the strong positive tilt of the isocurvature spectrum(niso > 1), this relative contribution increases towardssmaller scales. Since one of the observational constraintsthat we will consider in the following is represented by thenumber density of local galaxy clusters, it is interestingto estimate the isocurvature contribution at the charac-teristic cluster scales, ∼ 10 h−1Mpc. To this purpose weintroduce the quantity

3Note that this is equivalent to −2.64 < niso < −2.39 inthe usual notation, where niso = −3 is the scale invariantisocurvature perturbation.

6

FIG. 7. The ratio αclus/α between the relative isocurvaturecontribution α at the scale of clusters and at COBE scales,as a function of the tilt difference, niso − nad.

αclus/α = (kclus/khor)niso−nad = (300)niso−nad , (26)

which provides the isocurvature contribution at the scaleof galaxy clusters. We plot the cluster– to large–scaleratio of the isocurvature fraction in Fig. 7. It is apparentthat the enhancement of the isocurvature contributioncan be rather large, such that its effect is not negligibleat the cluster scale, even for a rather small large–scalecontribution.

We will discuss in Section V the implications of ourresults on the Linde–Mukhanov model. As for the cos-mological parameters Ωm, ΩΛ and H0, we always consid-ered flat models (Ωm +ΩΛ = 1), with 0.2 ≤ Ωm ≤ 1, andH0 ≡ 100 h = 50, 65, 80 km s−1 Mpc−1. As for the den-sity parameter contributed by baryons, we take the valueΩB = 0.019 h−2, which follows from the low deuteriumabundance, as determined by Burles & Tytler [28].

IV. OBSERVATIONAL CONSTRAINTS

In this section we test our model against the availablemicrowave background and large scale structure data,and we make predictions on the precision with whichfuture satellite experiments will determine the relevantparameters. To this purpose, we use the Fisher matrixtechnique to determine how well MAP and Planck satel-lites will constrain the isocurvature contribution to thetotal power spectrum and its spectral index.

A. The microwave background data

In the comparison of the CMB data with the modelpredictions, we performed a χ2 analysis, first applied toCMB data by Lineweaver et al. [29]. More precisely, we

computed the χ2, as a function of the cosmological pa-rameters ~λ, on the band–power estimates of the CMBdata, δT obs(n), and the model predictions, δT mod(~λ, n),given Nexp observed data points with their errors σ :

χ2(~λ) =Nexp∑n=1

[δT obs(n)− δT mod(~λ, n)

σ(n)

]2

. (27)

Evaluating the expression above, we used the 41 datapoints [30] listed in table I, and the corresponding win-dow functions. We chose not to introduce the recentpoints from MAT [31] and Python V [32] experimentsbecause the former does not yet include calibration er-rors and the results of the latter are still under discussion.The value of χ2 obtained is a function of the model pa-rameters ~λ.

For each choice of the parameters Ωm and H0, whichdescribe the Friedmann background, we compute the χ2

between model and data in the (niso, α) parameter space.In Table II we report the values of the parameters of thebest fit, the corresponding χ2

min value, and the probabil-ity of getting that χ2 value with present data if the modelconsidered is the real one.

For h = 0.5 a small contribution of the isocurvaturecomponent is desirable, especially for low Ωm universes.On the other hand, for h = 0.8 the addition of an isocur-vature contribution to the adiabatic mode doesn’t pro-vide a better fit to the data. In general, however, the al-lowed isocurvature fraction tends to be small (α < 0.01),while the isocurvature spectral index is not significantlyconstrained (see Fig. 4). In any case, the best fit to thedata is provided by the lowest niso considered.

B. Combining CMB and LSS constraints

In order to further constrain the parameter space ofallowed models, we will consider in this Section the con-straints coming from large scale structure observations.In particular, we will constrain the shape of the powerspectrum, by comparing to results from the analysis ofgalaxy clustering, and its amplitude by resorting to con-straints from the local abundance of rich galaxy clusters.

As for the shape of the galaxy power spectrum, dif-ferent determinations have been realized in the last fewyears, both for projected [26] and redshift [33] samples.Such analyses converge to indicate that the observedgalaxy power spectrum is well reproduced by an adia-batic CDM–like P (k), in a flat universe, with shape pa-rameter 0.2 <∼ Γ <∼ 0.3, for a scale invariant primordialspectrum [34].

According to the results of the previous section on theCMB constraints, the relative contribution of the isocur-vature component of the fluctuations is always rathersmall. As a result, the shape of the purely adiabaticspectrum is never significantly changed by the isocurva-ture component. For this reason, in order to implement

7

the constraint from the shape of the galaxy power spec-trum, we will simply require in the following that theadiabatic component of our mixed fluctuation spectrumhas a shape parameter Γ lying in the 0.2–0.3 range.

As for the amplitude of the power spectrum, a powerfulconstraint is represented by the number density of nearbygalaxy clusters. Since rich galaxy clusters involve a typi-cal mass of the order of 1015h−1M, their number densityis connected to the amplitude of density fluctuations atscales of about 10 h−1Mpc. Analytical approaches, basedon the method originally devised by Press & Schechter[35], show that the cluster abundance actually constrainsthe quantity σ8 = σ8Ωβ

m, where σ8 is the r.m.s. fluc-tuation amplitude within a top–hat sphere of 8 h−1Mpcand β ' 0.4–0.5, almost independent of the shape of thepower spectrum and of the presence of a cosmologicalconstant term [36].

Different analyses, based on the distributions of X–ray cluster luminosities, X–ray temperatures and veloc-ity dispersions of member galaxies, converge to valuesof σ8 in the range 0.5–0.6 [37]. For definiteness, in thefollowing we will use for our analysis the expression

σ8 = (0.55± 0.05)Ω−0.43+0.09Ωmm (28)

where the errors are intended to formally correspond toa 90% confidence level, while the Ωm dependence is thatprovided by Girardi et al. [38] for a flat Universe, withΩΛ = 1− Ωm.

We show in Figure 8 constraints on the niso−α param-eter space by combining results from LSS and CMB data.Each panel corresponds to a choice for the (Ωm, h) pa-rameters, among those reported in Table II, which agreeswith the measured shape of the galaxy power spectrum.

As for the constraints from the abundance of local clus-ters, the 90% confidence level region [see Eq. (28)] isshown by the shaded area. We note that, as niso in-creases, the contribution of the isocurvature fluctuationsat the cluster scale also increases, thus requiring a smallerα value to keep the power spectrum amplitude at the levelrequired by the cluster number density.

In order to establish confidence levels for model exclu-sion from the analysis of CMB anisotropies we considerthe quantity ∆χ2 = χ2 − χ2

min, where χ2min is the min-

imum value as reported in Table II. We assume the χ2

statistics for Ndof = 38 to be normally distributed withmean Ndof and r.m.s. scatter given by

√2Ndof . Accord-

ingly, all the models displayed in Figure 8 correspond toan acceptable value of χ2

min. The solid curve indicatesthe 90% c.l. upper limit on α, which corresponds to∆χ2 = 4.61 for two significant fitting parameters.

As a general result, it is interesting to note that largescale structure constraints significantly contribute to fur-ther restrict the range of the allowed parameter space.For instance, the two models with (Ωm, h) = (0.5, 0.65)and (0.4, 0.80) are now ruled out, since they can not sat-isfy at the same time both the CMB and local clusterabundance constraints, while the model with (Ωm, h) =

(0.5, 0.50) is constrained by the cluster abundance tohave α <∼ 0.0015.

It is worth reminding here that the constraint of Eq.(28) from the local cluster abundance has been derived inthe literature under the assumption of Gaussian statis-tics for the density fluctuations. On the other hand, thedensity fluctuations predicted by our model are given bya scale dependent superposition of a Gaussian adiabaticcomponents and of a non–Gaussian isocurvature compo-nent, whose probability density function (PDF) corre-sponds to a χ2 model with one degree of freedom. Al-though the extension of the Press–Schechter formalismto non–Gaussian statistics has been pursued by differ-ent authors [39], such attempts concentrated on scale–independent PDF models. In our case, we expect thatthe positive skewness of the χ2–distribution should easethe formation of galaxy clusters, for a fixed σ8, as a conse-quence of the broader high density tail in the PDF for theisocurvature component. Therefore, the net effect wouldgo in the direction of decreasing the required fluctuationamplitude at the cluster scale and, thus, to somewhatincrease the allowed α values. In any case, since theisocurvature component is always constrained to be rel-atively small even at the cluster scales (αclus

<∼ 0.15; cf.Fig. 7), we are confident that our assumption of Gaussianstatistics should be a sensible one.

C. The future CMB experiments

In this section we compute the estimates of the er-rors with which the future satellite experiments MAPand Planck will determine the isocurvature contributionto the CMB power spectrum. The aim here is to verifywhether the future CMB data alone will be able to con-strain small isocurvature contributions when other cos-mological parameters are constrained at the same time.

In order to provide such an estimate, we resort to theFisher information matrix approach [40]. When no po-larization is considered, the Fisher information matrix isdefined as

Fij =∑

l

∂Cl

∂λiCov−1 ∂Cl

∂λj, (29)

Cov =2

(2l + 1)fsky

(Cl + w−1el2σ2

b

)2

. (30)

Here fsky is the fraction of sky covered, σb is the Gaus-sian beamwidth (σb = θfwhm/

√8 ln 2, and θfwhm is the

full width at half maximum), λi is the set of parameterwhose errors have to be determined, and w contains theinformation on the detector resolution and sensitivity:

w =σ2

pixelΩpixel

T 20

, (31)

where Ωpixel ' θ2fwhm

.

8

FIG. 8. Constraints on the niso − α plane from cluster abundance and CMB anisotropies. Each panel corresponds to achoice for the (Ωm, h) parameters which satisfies the constraints on the shape parameter Γ. The corresponding Γ values foreach model are also reported. The shaded area correspond to the 90% c.l. from the cluster abundance, obtained according toeq.(28). The solid curves indicate the 90% c.l. upper limit on α from the CMB constraints (see text); oscillations in the shapeof these curves are due to limitations in the numerical precision.

In the case that also polarization is considered, theexpression for the Fisher and covariance matrices be-come more complicated, and we refer here to Ref. [41]for the explicit expression. We note that in the ex-pression above a perfect foreground subtraction is as-sumed, so that the estimates found should be consideredsomehow ideal. Also notice that the expression reportedhere strictly holds for Gaussian perturbations. Therefore,its application to the Linde–Mukhanov model should betaken with some caution, especially in the very large mul-tipole regime (i.e., small scales), where the non–Gaussianisocurvature contribution may be non negligible.

We estimate the expected errors on α and niso with andwithout polarization. In the calculation of the Fishermatrix, we considered as free parameters the Hubbleconstant h, the baryon abundance ΩBh2, the normaliza-tion C10, the reionization optical depth τ , the isocurva-ture/adiabatic ratio α, and the isocurvature spectral in-dex niso. Eqs. (30) and (29) are estimated by computingCl for a fiducial model with Ωm = 0.3 (fixed), h = 0.65,ΩBh2 = 0.019, C10 = C10COBE and τ = 0.05.

The precision in the determination of α and niso isreported in table IV for different fiducial values, and dif-ferent experiments. For both experiments, we combinedifferent channels in order to give the estimates on the er-rors. More precisely, we consider the 3 highest frequencychannels for MAP (40, 60, 90 GHz), and the 5 central fre-quency channels from Planck (70, 100 GHz channels fromLFI and the 100, 143, 217 GHz channels from HFI). The

values of the beamwidths and sensitivities for the twoexperiments are reported in table III, according to Refs.[1,2].

As a result, we find that the future satellite experi-ments will be able to constrain the spectral index niso

much better than the present data. Clearly, niso is bet-ter determined if a higher isocurvature component is al-lowed, and for a given α a higher fiducial isocurvatureanti-tilt ensures a smaller uncertainty in the parameterdetermination. Considering polarization is also useful inthis respect, and typically reduces the errors by almosta factor of 2. If the isocurvature contribution is as lowas indicated by our present analysis, niso will be anywaydetermined by Planck with an error of about 0.1.

As for the constraints on α, table IV shows that if theisocurvature contribution is quite large (α ' 0.015) bothMAP and Planck will be able to detect it; on the con-trary, if α is as low as the present data seem to suggest,only Planck (with polarization information included) willbe able to claim a detection, while MAP will only put anupper limit at the 0.002− 0.003 level. Note that the useof polarization data certainly helps in reducing the error,although we don’t find the big improvement claimed by[18]. This may occur because we keep the scalar to ten-sor ratio R fixed in this model, and therefore we are notaffected by the degeneracy of this parameter with α. Po-larization is useful in breaking the degeneracy betweenthe reionization parameter τ and the isocurvature con-tribution α and helps in reducing the errors especially in

9

the case of the Planck experiment.

V. DISCUSSION

In this paper we have investigated the consequencesof a CDM cosmology with mixed isocurvature and adia-batic initial conditions as prescribed by a generic Linde–Mukhanov inflationary model.

We showed how the total spectra are modified by theisocurvature contribution, as both the primordial spec-tral index for the isocurvature power spectrum, niso, andthe isocurvature fraction, α, are varied within their al-lowed ranges. In order to constrain the parameters ofthe model we resorted to available data on the CMBanisotropy, as well as on the large–scale structure con-straints from the shape of the galaxy power spectrumand the number density of nearby galaxy clusters. Ob-servational constraints from CMB and LSS data havebeen shown to provide complementary informations. Asa result, we found that the allowed isocurvature contri-bution at COBE scales is always very small, α <∼ 0.006.Therefore, our results generalizes to the niso > 1 case andstrengthen the conclusions reached in Ref. [16] based onLSS constraints alone.

We note that, even allowing for the strong positivetilt of the isocurvature component, the permitted isocur-vature contribution at the cluster scales is always small,with αclus

<∼ 0.15. However, this contribution is expectedto increase to ∼ 50% level when we consider smallerscales, <∼ 1 h−1Mpc, which are relevant for galaxy for-mation. Therefore, the resulting χ2 (positive skewness)non–Gaussian statistics contributed by the isocurvaturefluctuations can play a significant role to ease the galaxyformation at high redshift. The phenomenological impli-cations on galaxy formation of a small isocurvature con-tribution at the COBE scales remains to be investigatedin detail.

As for the determination of the isocurvature spectralindex niso, the best fit to the CMB data is always pro-vided by the lowest niso considered; although the limitsfrom LSS and CMB data in Fig. 8 show a mild depen-dence on niso. In this respect, future CMB experimentscan help to set more accurate limits on niso, with an op-timistic estimate of 3% and 1% (high niso and α) and amore realistic one of 20% and 8% for MAP and Planckrespectively.

Finally, we showed that only the Planck experimentwill be sensitive enough to detect a possible non vanishingvalue for the α parameter within the limits already setby present CMB and LSS data.

ACKNOWLEDGEMENTS

J.G.B. is supported by the Royal Society of London.E.P. is a CITA national fellow. The authors wish to ac-

knowledge SISSA for hosting all of them during differentphases of this work. The authors thank Andrei Linde,Andrew Liddle, and Douglas Scott for generous discus-sions.

[1] http://www.map.gsfc.nasa.gov[2] http://www.astro.estec.esa.nl/SA-general/

Projects/Planck/

[3] See, for example:J.R. Bond, G. Efstathiou and M. Tegmark, M.N.R.A.S.291 (1997) L33 [astro-ph/9702100]; L. Knox, Phys. Rev.D 52 (1995) 4307 [astro-ph/9504054]; G. Jungman, M.Kamionkowski, A. Kosowsky and D. Spergel, Phys. Rev.Lett. 76 (1996) 1007 [astro-ph/9507080]; M. Zaldar-riaga, D.N. Spergel and U. Seljak, Ap. J. 488 (1997)1 [astro-ph/9702157] and references therein.

[4] A.D. Linde, Phys. Lett. B 158 (1985) 375; L.A. Kof-man, Phys. Lett. B 173 (1985) 400; A.D. Linde and L.A.Kofman, Nucl. Phys. B 282 (1987) 555.

[5] G. Efstathiou and J.R. Bond, M.N.R.A.S. 218 (1986)103.

[6] H. Kodama and M. Sasaki, Int. J. Mod. Phys. A 2 (1987)491.

[7] W. Hu and N. Sugiyama, Phys. Rev. D 51 (1995) 2599[astro-ph/9411008].

[8] See, for example: D.S. Salopek, Phys. Rev. D 45 (1992)1139; L. Kofman and A.D. Linde, Nucl. Phys. B 282(1987) 555; L. Kofman, A.D. Linde and J. Einasto, Na-ture 326 (1987) 48; L. Kofman, Phys. Lett. B 173 (1986)400; M. Bucher and Y. Zhu, Phys. Rev. D 55 (1997) 7415[astro-ph/9610223].

[9] P.J.E. Peebles, Nature 327 (1987) 210; Ap. J. 315 (1987)L73; J.R. Bond and G. Efstathiou, M.N.R.A.S. 22 (1987)33; K. Gorski and J. Silk, Ap. J. 346 (1989) 1; J.Yokoyama and Y. Suto, Ap. J. 379 (1991) 427; T. Sugi-nohara and Y. Suto, Ap. J. 387 (1992) 431; R. Cen, J.P.Ostriker and P.J.E. Peebles, Ap. J. 415 (1993) 423; T.Chiba, N. Sugiyama and Y. Suto, Ap. J. 429 (1994) 427[astro-ph/9311012].

[10] W. Hu, E. Bunn and N. Sugiyama, Ap. J. 447 (1995) 59[astro-ph/9501034].

[11] J.A. Frieman and E. Gaztanaga, Ap. J. 521 (1999) L83[astro-ph/9903423].

[12] P.J.E. Peebles, Ap. J. 510 (1999) 523; Ap. J. 510 (1999)531.

[13] M. Bucher, K. Moodley and N. Turok, The general pri-mordial cosmic perturbation, astro-ph/9904231.

[14] S. Mollerach and S. Matarrese, Phys. Rev. D 45 (1992)1961; N. Deruelle, C. Gundlach and D. Langlois, Phys.Rev. D 46 (1992) 5337; A.A. Starobinsky and J.Yokoyama, Density fluctuations in Brans-Dicke infla-tion, gr-qc/9502002; J. Garcıa-Bellido and D. Wands,Phys. Rev. D 52 (1995) 6739 [gr-qc/9506050]; M. Sasakiand E.D. Stewart, Prog. Theor. Phys. 95 (1996) 71[astro-ph/9507001]; J. Garcıa-Bellido and D. Wands,

10

Phys. Rev. D 53 (1996) 5437 [astro-ph/9511029]; D.Langlois, Correlated adiabatic and isocurvature perturba-tions from double inflation, astro-ph/9906080.

[15] R. Stompor, A.J. Banday and K.M. Gorski, Ap. J. 463(1996) 8 [astro-ph/9511087].

[16] M. Kawasaki, N. Sugiyama and T. Yanagida, Phys. Rev.D 54 (1996) 2442 [hep-ph/9512368]; T. Kanazawa, M.Kawasaki, N. Sugiyama and T. Yanagida, Prog. Theor.Phys. 102 (1999) 71 [astro-ph/9805102].

[17] A.D. Linde and V. Mukhanov, Phys. Rev. D 56 (1997)535 [astro-ph/9610219].

[18] K. Enqvist and H. Kurki–Suonio, Constrain-ing isocurvature fluctuations with the Planck Surveyor,astro-ph/9907221.

[19] U. Seljak and M. Zaldarriaga, Ap. J. 469 (1996) 437[astro-ph/9603033].

[20] CMBFAST webpage: http://www.sns.ias.edu/∼matiasz/CMBFAST/cmbfast.html

[21] C.L. Bennett et al., Ap. J. 464 (1996) L1[astro-ph/9601067].

[22] J.R. Bond, in Cosmology and Large Scale Structure, LesHouches Summer School Course LX, R. Schaeffer ed.,Elsevier Science Press, Amsterdam, 1996.

[23] E.F. Bunn and M. White, Ap. J. 480 (1997) 6[astro-ph/9706060].

[24] J.M. Bardeen, J.R. Bond, N. Kaiser and A.S. Szalay, Ap.J. 304 (1986) 15.

[25] N. Sugiyama, Ap. J. Suppl. 100 (1995) 281[astro-ph/9412025].

[26] C.M. Baugh and G. Efstathiou, M.N.R.A.S. 265 (1993)145; E. Gaztanaga and C.M. Baugh, M.N.R.A.S. 294(1998) 229 [astro-ph/9704246].

[27] J.R. Bond, A.H. Jaffe, L. Knox, Radical compression ofcosmic microwave background data astro-ph/9808264.

[28] S. Burles and D. Tytler, Ap. J. 507 (1998) 732[astro-ph/9712109].

[29] C.H. Lineweaver, D. Barbosa, A. Blanchard and J.G.Bartlett, A. and A., 322 (1997) 365 [astro-ph/9610133].

[30] M. Tegmark and A. Hamilton, Uncorrelated measure-ments of the CMB power spectrum, astro-ph/9702019;G.S. Tucker, H.P. Gush, M. Halpern, I. Shinkoda, Ap.J. 475 (1997) L73; K. Ganga, L. Page, E. S. Cheng, S.Meyer, Ap. J. 432 (1994) L15; S.T. Tanaka et al., Ap. J.468 (1996) L81; P. de Bernardis et al., Ap. J. 422 (1994)L33; A. de Oliveira-Costa et al., Ap. J. 509 (1998) L77;S. Masi et al., Ap. J. 463 (1996) L47; C.B. Netterfieldet al., Ap. J. 474 (1997) 47; S.T. Tanaka et al., Ap.J. 468 (1996) L81; E. Leitch et al., A measurement ofanisotropy in the cosmic microwave background on 7-22arcminute scales, astro-ph/9807312 (1998); J.C. Bakeret al., Detection of cosmic microwave background struc-ture in a second field with the Cosmic Anisotropy Tele-scope, astro-ph/9904415; J.O. Gundersen et al., Ap. J.433 (1995) L57; B. Femenia, R. Rebolo, C.M. Gutier-rez, M. Limon, L. Piccirillo, Ap. J. 498 (1998) 117; S.Hancock et al., M.N.R.A.S. 289 (1997) 505; L. Piccirilloand P. Calisse, Ap. J. 411 (1993) 529; J.O. Gundersen etal., Ap. J. 433 (1995) L57; S.R. Platt et al., Ap. J. 475(1997) L1; P.F.S. Scott et al., Ap. J. 461 (1996) L1; E.S.Cheng et al., Ap. J. 456 (1996) L71; E.S. Cheng et al.,

Ap. J. 488 (1997) L59; S.T. Tanaka et al., Ap. J. 468(1996) L81.

[31] E. Torbet et al., Ap. J. 521 (1999) L79[astro-ph/9905100].

[32] K. Coble et al., Anisotropy in the cosmic microwavebackground at degree angular scales: Python V results,astro-ph/9902195.

[33] H. Lin et al., Ap. J. 471 (1996) 617 [astro-ph/9606055];H. Tadros et al., M.N.R.A.S. 305 (1999) 527[astro-ph/9901351].

[34] J.A. Peacock and S.J. Dodds, M.N.R.A.S. 267 (1994)1020 [astro-ph/9311057]; A.R. Liddle, D.H. Lyth, R.H.Schaefer, Q. Shafi and P.T.P. Viana, M.N.R.A.S. 281(1996) 531 [astro-ph/9511057].

[35] W.H. Press and P. Schechter, Ap. J. 187 (1974) 425;C.G. Lacey and S. Cole, M.N.R.A.S. 271 (1994) 676[astro-ph/9402069].

[36] S.D.M. White, G. Efstathiou and C.S. Frenk,M.N.R.A.S. 262 (1993) 1023.

[37] V.R. Eke, S. Cole and C.S. Frenk, M.N.R.A.S. 282(1996) 263 [astro-ph/9601088]; T. Kitayama and Y.Suto, Ap. J. 469 (1996) 480 [astro-ph/9704141]; M.Markevitch, Ap. J. 504 (1998) 27; S. Borgani, P.Rosati, P. Tozzi and C. Norman, Ap. J. 517 (1999)40 [astro-ph/9901017]; P.T.P. Viana and A.R. Liddle,M.N.R.A.S. 303 (1999) 535 [astro-ph/9803244].

[38] M. Girardi, S. Borgani, G. Giuricin, F. Mardirossian andM. Mezzetti, Ap. J. 506 (1998) 45 [astro-ph/9804188].

[39] S. Colafrancesco, F. Lucchin and S. Matarrese, Ap. J.345 (1989) 3; W.A. Chiu, J.P. Ostriker and M.A. Strauss,Ap. J. 494 (1998) 479 [astro-ph/9708250]; J. Robinsonand J.E. Baker, Evolution of the cluster abundance innon-gaussian models, astro-ph/9905098; K. Koyama, J.Soda and A. Taruya, Constraints on a non-gaussian (χ2

m)CDM model, astro-ph/9903027.

[40] M. Tegmark, A. Taylor and A. Heavens, Ap. J. 480(1997) 22 [astro-ph/9603021].

[41] M. Zaldarriaga, D.N. Spergel, U. Seljak, Ap. J. 488(1997) 1 [astro-ph/9702157].

11

Experiment ∆Tl ± σ(µK) leff reference

COBE 8.5+16−8.5 2.1 Tegmark & Hamilton (1997)

COBE 28.0+7.4−10.4 3.1 Tegmark & Hamilton (1997)





COBE 26.1+4.45.3 14.3 Tegmark & Hamilton (1997)


SASK 49.0+8.0−5.0 86 Netterfield et al (1997)




SASK 69.0+19.028.0 348 Netterfield et al (1997)

CAT 50.8+15.4−15.4 396 Scott et al (1996)

CAT 49.0+16.9−16.9 608 Scott et al (1996)

CAT 57.3+10.9−13.6 415 Baker et al (1998)

FIRS 29.4+7.8−7.7 10 Ganga et al (1994)

TENERIFE 34.1+12.5−12.5 20 Hancock et al (1997)

SP91 30.2+8.9−5.5 57 Gundersen et al (1995)

SP94 36.3+13.6−6.1 57 Gundersen et al (1995)

BAM 48.4+16.5−16.5 74 Tucker et al (1997)

ARGO 39.1+8.7−8.7 95 de Bernardis et al (1994)

ARGO 46.8+9.5−12.1 95 Masi et al (1996)

MAX-GUM 54.5+16.4−10.9 145 Tanaka et al (1996)

MAX-ID 46.3+21.8−13.6 145 Tanaka et al (1996)

MAX-SH 49.1+21.8−16.4 145 Tanaka et al (1996)

MAX-HR 32.7+10.9−8.2 145 Tanaka et al (1996)

MAX-PH 51.8+19.1−10.9 145 Tanaka et al (1996)

PYTHON1 54.0+14.0−12.0 92 Platt et al (1996)

PYTHON2 58.0+15.0−13.0 177 Platt et al (1996)

IAC 112.0+65.0−60.0 33 Femenia et al (1998)

IAC 55.0+27.0−22.0 53 Femenia et al (1998)

IAB 94.5+41.8−41.8 125 Piccirillo & Calisse (93)

MSAM 62.0+21.7−21.7 143 Cheng et al (1996)

MSAM 60.4+20.1−20.1 249 Cheng et al (1996)

MSAM 50.0+16.0−11.0 160 Cheng et al (1997)

MSAM 65.0+18.0−13.0 270 Cheng et al (1997)

QMAP 47.0+6.−7. 80 deOliveira–Costa et al (98)



OVRO 56.0+8.5−6.6 589 Leitch et al (1998)

TABLE I. Data points used in the χ2 analysis. First col-umn is the experiment, second column is the experimentalvalue with its error, third column is the effective multipolenumber, and fourth column is the reference paper.

Ωm niso α χmin P (χ < χmin)

0.2 1.359 0.006 21.6 0.0150.3 1.359 0.003 21.9 0.0170.4 1.359 0.001 22.7 0.0230.5 1.359 0 23.5 0.0320.6 1.359 0 24.8 0.0480.7 1.359 0 26.4 0.0780.8 1.359 0 28.3 0.1260.9 1.359 0 30.5 0.2001 1.359 0 33.5 0.323

0.2 1.359 0.001 24.4 0.0430.3 1.359 0 26.4 0.080.4 1.359 0 29.2 0.150.5 1.359 0 32.7 0.290.6 1.359 0 37.4 0.500.7 1.359 0 43.1 0.740.8 1.359 0 49.2 0.890.9 1.359 0 55.5 0.971 1.359 0 61.8 0.99

0.2 1.359 0 30.2 0.190.3 1.359 0 36.0 0.440.4 1.359 0 44.1 0.770.5 1.359 0 52.8 0.940.6 1.359 0 61.5 0.990.7 1.359 0 69.8 0.990.8 1.359 0 77.8 0.990.9 1.359 0 85.4 0.991 1.359 0 92.6 0.99

TABLE II. Results of the χ2 analysis for different models(h = 0.50, 0.65, 0.80 from top to bottom). Column 1: thematter density parameter. Column 2 and 3: the best fit valuesfor niso and α respectively. Column 4: the correspondingχ2

min value. Column 5: the probability of getting a smallerχ2, assuming that this is the real model. We considered here41 experimental points and 3 free parameters (α, niso, andthe normalization), which corresponds to Ndof = 38 degreesof freedom.

Experiment frequency (GHz) θfwhm σp (µK) σpolp (µK)

MAP 40 0.47 35 49.3MAP 60 0.35 35 49.3MAP 90 0.21 35 49.3

Planck-LFI 70 14′ 9.8 13.8Planck-LFI 100 10′ 11.72 16.5Planck-HFI 100 10.7′ 4.63 –Planck-HFI 143 8.0′ 5.45 10.2Planck-HFI 217 5.5′ 11.7 26.2

TABLE III. Beamwidths and sensitivities of the satellitesexperiments MAP and Planck.

12

fiducial fiducial MAP MAP Planck Planckα niso δα δniso δα δniso

0.002 1.439 0.0035 0.315 0.0024 0.2090.002 1.572 0.0022 0.195 0.0016 0.1390.008 1.439 0.0044 0.094 0.0031 0.0640.008 1.572 0.0030 0.062 0.0023 0.0470.015 1.439 0.0054 0.059 0.0040 0.0420.015 1.572 0.0039 0.041 0.0031 0.031

0.002 1.439 0.0025 0.297 0.0013 0.1260.002 1.572 0.0015 0.172 0.0008 0.0730.008 1.439 0.0031 0.091 0.0017 0.0390.008 1.572 0.0020 0.056 0.0010 0.0240.015 1.439 0.0038 0.059 0.0020 0.0250.015 1.572 0.0026 0.038 0.0014 0.016

TABLE IV. Estimation of the isocurvature mode parame-ters with the future satellite experiments MAP and Planck.Top: polarization not considered. Bottom: polarization con-sidered. The first two columns indicate the fiducial valueconsidered, and the others indicate the estimated errors inthe parameters.

13

microwave background anisotropies and large scale ...by linde and mukhanov. we provide constraints...

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